Gödel and the End of Physics Stephen Hawking

Notes

Stephen Hawking, the Cambridge University physicist famous for his theories on black holes and his best-selling books about the universe, presented a public lecture on March 8 at Texas A&M University.

Hawking was visiting Texas A&M as part of a month-long physics conference to inaugurate the university's George P. and Cynthia W. Mitchell Institute for Fundamental Physics, established with an endowment from the well-known Texas businessman and his wife, both of The Woodlands.

"The chance to hear the world's best-known living scientist discuss scientific wonders of the universe is a rare gift for us all," said H. Joseph Newton, dean of Texas A&M's College of Science. "This is the kind of event that will define the Mitchell Institute as a world-class forum in theoretical physics."

"This is the first time that we have filled a classroom for physics!", said Dr. Peter McIntyre, professor of physics and Texas A&M. Dr. McIntyre then introduced Dr. Christopher Pope
who was a student of
Hawking. Dr. Pope pointed out some of the highlights of Hawking's
career including his appearance in Star Trek: The Next Generation and
The Simpsons.
Hawking, whose best-selling books A Brief History of Time and The Universe in a Nutshell have sold millions of copies around the world, is renowned as a scientist with the uncommon ability to communicate complex science in a way that touches people.
At 4 p.m. on March 8, he presented "Gödel and the End of Physics" to a sold-out crowd in Rudder Auditorium, on Texas A&M's College Station campus.

Hawking is certainly the most famous physicist in history who has not won the Nobel Prize. This has puzzled people. They automatically assume he has won the Nobel Prize. He has not yet. This is because the Swedish Royal Academy demands that an award-winning discovery must be supported by verifiable experimental or observational evidence. Hawking's work, to date, remains unproved. The mathematics of his theory, however, are certainly beautiful and elegant. Science is just beginning to verify the existence of black holes, let alone verify "Hawking radiation" or any of his more radical theoretical proposals.

Transcript

Can you hear me?

I'm afraid that my accent is not Texan.

In this talk, I want to ask how far can we go, in our search for
understanding and knowledge. Will we ever find a complete form of the laws
of nature. By a complete form, I mean a set of rules, that in principle at
least, enable us to predict the future to an arbitrary accuracy, knowing
the state of the universe at one time. A qualitative understanding of the
laws, has been the aim of philosophers and scientists, from Aristotle
onwards. But it was Newton's Principia Mathematica in 1687, containing his
theory of universal gravitation, that made the laws quantitative and
precise. This led to the idea of scientific determinism, which seems first
to have been expressed by Laplace. If at one time, one knew the positions
and velocities of all the particles in the universe, the laws of science
should enable us to calculate their positions and velocities, at any other
time, past or future. The laws may or may not have been ordained by God,
but scientific determinism asserts that he does not intervene, to break
them.

At first, it seemed that these hopes for a complete determinism would
be dashed, by the discovery early in the 20th century, that events like
the decay of radio active atoms, seemed to take place at random. It was as
if God was playing dice, in Einstein's phrase. But science snatched
victory from the jaws of defeat, by moving the goal posts, and redefining
what is meant by a complete knowledge of the universe. It was a stroke of
brilliance, whose philosophical implications have still not been fully
appreciated. Much of the credit belongs to Paul Dirac, my predecessor but
one in the Lucasian chair, though it wasn't motorized in his time.
Dirac
showed how the work of Erwin Schrödinger and Werner Heisenberg, could be
combined in new picture of reality, called quantum theory. In quantum
theory, a particle is not characterized by two quantities, its position
and its velocity, as in classical Newtonian theory. Instead it is described
by a single quantity, the wave function. The size of the wave function at
a point, gives the probability that the particle will be found at that
point, and the rate at which the wave function changes from point to
point, gives the probability of different velocities. One can have a wave
function that is sharply peaked at a point. This corresponds to a state in
which there is little uncertainty in the position of the particle. However,
the wave function varies rapidly, so there is a lot of uncertainty in the
velocity. Similarly, a long chain of waves has a large uncertainty in
position, but a small uncertainty in velocity. One can have a well defined
position, or a well defined velocity, but not both.

This would seem to make complete determinism impossible. If one can't
accurately define both the positions, and the velocities, of particles at
one time, how can one predict what they will be in the future. It is like
weather forecasting. The forecasters don't have an accurate knowledge of
the atmosphere at one time. Just a few measurements at ground level, and
what can be learnt from satellite photographs. That’s why weather forecasts
are so unreliable. However, in quantum theory, it turns out one doesn't
need to know both the positions, and the velocities. If one knew the laws
of physics, and the wave function at one time, then something called the
Schrödinger equation, would tell one how fast the wave function was
changing with time. This would allow one to calculate the wave function at
any other time. One can therefore claim that there is still determinism,
but it is a determinism on a reduced level Instead of being able
accurately to predict two quantities, position and velocity, one can
predict only a single quantity, the wave function. We have re-defined
determinism, to be just half of what Laplace thought it was. Some people
have tried to connect the unpredictability of the other half, with
consciousness, or the intervention of supernatural beings. But it is
difficult to make either case, for something that is completely random.

In order to calculate how the wave function develops in time, one needs
the quantum laws that govern the universe. So how well do we know these
laws. As Dirac remarked, Maxwell's equations of light, and the
relativistic wave equation, which he was too modest to call the Dirac
equation, govern most of physics, and all of chemistry and biology. So in
principle, we ought to be able to predict human behavior, though I can't
say I have had much success myself. The trouble is that the human brain
contains far too many particles, for us to be able to solve the
equations. But it is comforting to think we might be able to predict the
nematode worm, even if we can't quite figure out humans. Quantum theory,
and the Maxwell and Dirac equations, indeed govern much of our life, but
there are two important areas beyond their scope. One is the nuclear
forces. The other is gravity. The nuclear forces are responsible for the
Sun shining, and the formation of the elements, including the carbon and
oxygen of which we are made. And gravity caused the formation of stars and
planets, and indeed, of the universe itself. So it is important to bring
them into the scheme.

The so called weak nuclear forces, have been unified with the Maxwell
equations, by Abdus Salahm and Stephen Weinberg, in what is known as, the
Electro weak theory. The predictions of this theory have been confirmed by
experiment, and the authors rewarded with Nobel prizes. The remaining
nuclear forces, the so called strong forces, have not yet been
successfully unified with the electro weak forces, in an observationally
tested scheme. Instead, they seem to be described by a similar but
separate theory, called QCD. It is not clear who, if anyone, should get a
Nobel prize for QCD, but David Gross and Gerardus 't Hooft, share credit
for showing the theory gets simpler at high energies. I had quite a job to
get my speech synthesizer to pronounce Gerardus surname. It wasn't
familiar with apostrophe t. The electro weak theory, and QCD, together
constitute the so called Standard Model of particle physics, which aims to
describe everything except gravity.

The standard model seems to be adequate for all practical purposes, at
least for the next hundred years. But practical or economic reasons, have
never been the driving force in our search for a complete theory of the
universe. No one working on the basic theory, from Galileo onward, has
carried out their research to make money, though Dirac would have made a
fortune if he had patented the Dirac equation. He would have had a royalty
on every television, walkman, video game and computer.

The real reason we are seeking a complete theory, is that we want to
understand the universe, and feel we are not just the victims of dark and
mysterious forces. If we understand the universe, then we control it, in a
sense. The standard model is clearly unsatisfactory in this respect. First
of all, it is ugly and ad hoc. The particles are grouped in an apparently
arbitrary way, and the standard model depends on 24 numbers, whose values
can not be deduced from first principles, but which have to be chosen to
fit the observations. What understanding is there in that? Can it be
Nature's last word. The second failing of the standard model, is that it
does not include gravity. Instead, gravity has to be described by
Einstein's General Theory of Relativity. General relativity, is not a
quantum theory, unlike the laws that govern everything else in the
universe. Although it is not consistent to use the non quantum general
relativity, with the quantum standard model, this has no practical
significance at the present stage of the universe, because gravitational
fields are so weak. However, in the very early universe, gravitational
fields would have been much stronger, and quantum gravity would have been
significant. Indeed, we have evidence that quantum uncertainty in the
early universe, made some regions slightly more or less dense, than the
otherwise uniform background. We can see this in small differences in the
background of microwave radiation from different directions. The hotter,
denser regions will condense out of the expansion as galaxies, stars and
planets. All the structures in the universe, including ourselves, can be
traced back to quantum effects in the very early stages. It is therefore
essential to have a fully consistent quantum theory of gravity, if we are
to understand the universe.

Constructing a quantum theory of gravity, has been the outstanding
problem in theoretical physics, for the last 30 years. It is much, much
more difficult than the quantum theories of the strong and electro weak
forces. These propagate in a fixed background of space and time. One can
define the wave function, and use the Schrödinger equation to evolve it
in time. But according to general relativity, gravity is space and time. So
how can the wave function for gravity, evolve in time. And anyway, what
does one mean by the wave function for gravity. It turns out that, in a
formal sense, one can define a wave function, and a Schrödinger like
equation for gravity, but that they are of little use in actual
calculations.

Instead, the usual approach is to regard the quantum spacetime, as a
small perturbation of some background spacetime, generally flat space. The
perturbations can then be treated as quantum fields, like the electro weak
and QCD fields, propagating through the background spacetime. In
calculations of perturbations, there is generally some quantity, called
the effective coupling, which measures how much of an extra perturbation,
a given perturbation generates. If the coupling is small, a small
perturbation, creates a smaller correction, which gives an even smaller
second correction, and so on. Perturbation theory works, and can be used to
calculate to any degree of accuracy. An example is your bank account. The
interest on the account, is a small perturbation. A very small perturbation
if you are with one of the big banks). The interest is compound. That is,
there is interest on the interest, and interest on the interest on the
interest. However, the amounts are tiny. To a good approximation, the money
in your account, is what you put there. On the other hand, if the coupling
is high, a perturbation generates a larger perturbation, which then
generates an even larger perturbation. An example would be borrowing money
from loan sharks. The interest can be more than you borrowed, and then you
pay interest on that. It is disastrous.

With gravity, the effective coupling is the energy or mass of the
perturbation, because this determines how much it warps spacetime, and so
creates a further perturbation. However, in quantum theory, quantities
like the electric field, or the geometry of spacetime, don't have definite
values, but have what are called, quantum fluctuations. These fluctuations
have energy. In fact, they have an infinite amount of energy, because there
are fluctuations on all length scales, no matter how small. Thus treating
quantum gravity as a perturbation of flat space, doesn't work well,
because the perturbations are strongly coupled.

Supergravity was invented in 1976 to solve, or at least improve, the
energy problem. It is a combination of general relativity with other
fields, such that that each species of particle, has a super partner
species. The energy of the quantum fluctuations of one partner is
positive, and the other negative, so they tend to cancel. It was hoped the
infinite positive and negative energies would cancel completely, leaving
only a finite remainder. In this case, a perturbation treatment would
work, because the effective coupling would be weak. However in 1985,
people suddenly lost confidence that the infinities would cancel. This was
not because anyone had shown that they definitely didn't cancel. It was
reckoned it would take a good graduate student, 300 years to do the
calculation, and how would one know they hadn't made a mistake on page
two. Rather it was because Ed Witten declared that string theory, was the
true quantum theory of gravity, and supergravity was just an
approximation, valid when particle energies are low, which in practice,
they always are. In string theory, gravity is not thought of as the
warping of spacetime. Instead, it is given by string diagrams, networks of
pipes that represent little loops of string, propagating through flat
spacetime. The effective coupling, that gives the strength of the
junctions where three pipes meet, is not the energy, as it is in
supergravity. Instead it is given by what is called, the dilaton, a field
that has not been observed. If the dilaton had a low value, the effective
coupling would be weak, and string theory, would be a good quantum
theory. But it is no earthly use for practical purposes.

In the years since 1985, we have realized that both supergravity and
string theory, belong to a larger structure, known as M theory. Why it
should be called M Theory, is completely obscure. M theory, is not a
theory in the usual sense. Rather it is a collection of theories, that look
very different, but which describe the same physical situation. These
theories are related by mappings, or correspondences, called dualities,
which imply that they are all reflections of the same underlying theory.
Each theory in the collection, works well in the limit, like low energy,
or low dilaton, in which its effective coupling is small, but breaks down
when the coupling is large. This means that none of the theories, can
predict the future of the universe, to arbitrary accuracy. For that, one
would need a single formulation of M-theory, that would work in all
situations.

Up to now, most people have implicitly assumed that there is an
ultimate theory, that we will eventually discover. Indeed, I myself have
suggested we might find it quite soon. However, M-theory has made me
wonder if this is true. Maybe it is not possible to formulate the theory of
the universe in a finite number of statements. This is very reminiscent of
Gödel’s theorem. This says that any finite system of axioms, is not
sufficient to prove every result in mathematics.

Gödel’s theorem is proved using statements that refer to
themselves. Such statements can lead to paradoxes. An example is, this
statement is false. If the statement is true, it is false. And if the
statement is false, it is true. Another example is, the barber of Corfu
shaves every man who does not shave himself. Who shaves the barber? If he
shaves himself, then he doesn't, and if he doesn't, then he does. Gödel
went to great lengths to avoid such paradoxes, by carefully distinguishing
between mathematics, like 2+2 =4,and meta mathematics, or statements about
mathematics, such as mathematics is cool, or mathematics is consistent.
that is why his paper is so difficult to read. But the idea is quite
simple. First Gödel showed that each mathematical formula, like 2+2=4,
can be given a unique number, the Gödel number. The Gödel number of
2+2=4, is *. Second, the meta mathematical statement, the sequence of
formulas A, is a proof of the formula B, can be expressed as an
arithmetical relation between the Gödel numbers for A- and B. Thus meta
mathematics can be mapped into arithmetic, though I'm not sure how you
translate the meta mathematical statement, 'mathematics is cool'. Third
and last, consider the self referring Gödel statement, G. This is, the
statement G can not be demonstrated from the axioms of mathematics.
Suppose that G could be demonstrated. Then the axioms must be inconsistent,
because one could both demonstrate G, and show that it can not be
demonstrated. On the other hand, if G can't be demonstrated, then G is
true. By the mapping into numbers, it corresponds to a true relation
between numbers, but one which can not be deduced from the axioms. Thus
mathematics is either inconsistent, or incomplete. The smart money, is on
incomplete.

What is the relation between Gödel’s theorem, and whether we can
formulate the theory of the universe, in terms of a finite number of
principles. One connection is obvious. According to the positivist
philosophy of science, a physical theory, is a mathematical model. So if
there are mathematical results that can not be proved, there are physical
problems that can not be predicted. One example might be the Golbach
conjecture. Given an even number of wood blocks, can you always divide them
into two piles, each of which can not be arranged in a rectangle. That is,
it contains a prime number of blocks.

Although this is incompleteness of sort, it is not the kind of
unpredictability I mean. Given a specific number of blocks, one can
determine with a finite number of trials, whether they can be divided into
two primes. But I think that quantum theory and gravity together,
introduces a new element into the discussion, that wasn't present with
classical Newtonian theory. In the standard positivist approach to the
philosophy of science, physical theories live rent free in a Platonic
heaven of ideal mathematical models. That is, a model can be arbitrarily
detailed, and can contain an arbitrary amount of information, without
affecting the universes they describe. But we are not angels, who view the
universe from the outside. Instead, we and our models, are both part of the
universe we are describing. Thus a physical theory, is self referencing,
like in Gödel’s theorem. One might therefore expect it to be either
inconsistent, or incomplete. The theories we have so far, are ~both
inconsistent, and incomplete.

Quantum gravity is essential to the argument. The information in the
model, can be represented by an arrangement of particles. According to
quantum theory, a particle in a region of a given size, has a certain
minimum amount of energy. Thus, as I said earlier, models don't live rent
free. They cost energy. By Einstein’s famous equation, E = mc squared, energy
is equivalent to mass. And mass causes systems to collapse under gravity.
It is like getting too many books together in a library. The floor would
give way, and create a black hole that would swallow the information.
Remarkably enough, Jacob Bekenstein and I, found that the amount of
information in a black hole, is proportional to the area of the boundary
of the hole, rather than the volume of the hole, as one might have
expected. The black hole limit on the concentration of information, is
fundamental, but it has not been properly incorporated into any of the
formulations of M theory that we have so far. They all assume that one can
define the wave function at each point of space. But that would be an
infinite density of information, which is not allowed. On the other hand,
if one can't define the wave function point wise, one can't predict the
future to arbitrary accuracy, even in the reduced determinism of quantum
theory. What we need, is a formulation of M theory, that takes account of
the black hole information limit. But then our experience with supergravity
and string theory, and the analogy of Gödel’s theorem, suggest that even
this formulation, will be incomplete.

Some people will be very disappointed if there is not an ultimate
theory, that can be formulated as a finite number of principles. I used to
belong to that camp, but I have changed my mind. I'm now glad that our
search for understanding will never come to an end, and that we will
always have the challenge of new discovery. Without it, we would stagnate.
Gödel’s theorem ensured there would always be a job for mathematicians. I
think M theory will do the same for physicists. I'm sure Dirac would have
approved.